Properties

Label 5600.2.a.by
Level $5600$
Weight $2$
Character orbit 5600.a
Self dual yes
Analytic conductor $44.716$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5600,2,Mod(1,5600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5600.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7162251319\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.29935424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 18x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{7} + ( - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{7} + ( - \beta_{2} + \beta_1 + 1) q^{9} + \beta_{4} q^{11} + (\beta_{4} - \beta_{3}) q^{13} + ( - \beta_{5} + \beta_{3}) q^{17} + (\beta_{5} + \beta_{4}) q^{19} + \beta_1 q^{21} + ( - 2 \beta_1 + 2) q^{23} + (\beta_{2} - \beta_1 - 2) q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + 2 \beta_{3} q^{31} + ( - \beta_{4} - 2 \beta_{3}) q^{33} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{37} + ( - \beta_{5} - \beta_{3}) q^{39} + (2 \beta_1 + 2) q^{41} + ( - 2 \beta_1 - 4) q^{43} + ( - 3 \beta_{2} + \beta_1 + 2) q^{47} + q^{49} + ( - \beta_{4} - 4 \beta_{3}) q^{51} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{53} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{57} + ( - \beta_{5} - \beta_{4}) q^{59} + (\beta_{2} - 2 \beta_1 + 4) q^{61} + (\beta_{2} - \beta_1 - 1) q^{63} + (2 \beta_{2} - 2 \beta_1 - 4) q^{67} + ( - 2 \beta_{2} + 8) q^{69} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{71} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{73} - \beta_{4} q^{77} + (\beta_{5} + \beta_{3}) q^{79} + (2 \beta_{2} + 2 \beta_1 - 1) q^{81} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{83} + ( - \beta_{2} - 5 \beta_1 + 6) q^{87} + 6 q^{89} + ( - \beta_{4} + \beta_{3}) q^{91} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{93} + (\beta_{5} + 2 \beta_{4} - \beta_{3}) q^{97} + ( - 2 \beta_{5} + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} + 2 q^{21} + 8 q^{23} - 14 q^{27} + 22 q^{29} + 16 q^{41} - 28 q^{43} + 14 q^{47} + 6 q^{49} + 20 q^{61} - 8 q^{63} - 28 q^{67} + 48 q^{69} - 2 q^{81} - 16 q^{83} + 26 q^{87} + 36 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 18x^{2} - 14x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + \nu^{3} + 6\nu^{2} - 3\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 8\nu^{2} + 5\nu - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} + 8\nu^{3} - 4\nu^{2} - 14\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 3\beta_{4} - 2\beta_{3} + \beta_{2} - 4\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{5} + 9\beta_{4} + \beta_{3} + 14\beta_{2} - 14\beta _1 + 74 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35\beta_{5} + 39\beta_{4} - 21\beta_{3} + 22\beta_{2} - 42\beta _1 + 122 ) / 4 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.178681
−1.67709
−1.97020
2.64856
2.36682
0.810591
0 −2.85577 0 0 0 −1.00000 0 5.15544 0
1.2 0 −2.85577 0 0 0 −1.00000 0 5.15544 0
1.3 0 −0.321637 0 0 0 −1.00000 0 −2.89655 0
1.4 0 −0.321637 0 0 0 −1.00000 0 −2.89655 0
1.5 0 2.17741 0 0 0 −1.00000 0 1.74111 0
1.6 0 2.17741 0 0 0 −1.00000 0 1.74111 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5600.2.a.by 6
4.b odd 2 1 5600.2.a.bz 6
5.b even 2 1 5600.2.a.bz 6
5.c odd 4 2 1120.2.g.d 12
20.d odd 2 1 inner 5600.2.a.by 6
20.e even 4 2 1120.2.g.d 12
40.i odd 4 2 2240.2.g.p 12
40.k even 4 2 2240.2.g.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.d 12 5.c odd 4 2
1120.2.g.d 12 20.e even 4 2
2240.2.g.p 12 40.i odd 4 2
2240.2.g.p 12 40.k even 4 2
5600.2.a.by 6 1.a even 1 1 trivial
5600.2.a.by 6 20.d odd 2 1 inner
5600.2.a.bz 6 4.b odd 2 1
5600.2.a.bz 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5600))\):

\( T_{3}^{3} + T_{3}^{2} - 6T_{3} - 2 \) Copy content Toggle raw display
\( T_{11}^{6} - 45T_{11}^{4} + 592T_{11}^{2} - 1856 \) Copy content Toggle raw display
\( T_{13}^{6} - 65T_{13}^{4} + 1056T_{13}^{2} - 116 \) Copy content Toggle raw display
\( T_{19}^{6} - 104T_{19}^{4} + 2708T_{19}^{2} - 4176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} + T^{2} - 6 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 45 T^{4} + 592 T^{2} + \cdots - 1856 \) Copy content Toggle raw display
$13$ \( T^{6} - 65 T^{4} + 1056 T^{2} + \cdots - 116 \) Copy content Toggle raw display
$17$ \( T^{6} - 109 T^{4} + 3608 T^{2} + \cdots - 37584 \) Copy content Toggle raw display
$19$ \( T^{6} - 104 T^{4} + 2708 T^{2} + \cdots - 4176 \) Copy content Toggle raw display
$23$ \( (T^{3} - 4 T^{2} - 20 T + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 11 T^{2} + 16 T + 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 128 T^{4} + 4416 T^{2} + \cdots - 29696 \) Copy content Toggle raw display
$37$ \( T^{6} - 168 T^{4} + 8080 T^{2} + \cdots - 118784 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} - 4 T + 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 14 T^{2} + 40 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 7 T^{2} - 104 T + 776)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 204 T^{4} + 8176 T^{2} + \cdots - 1856 \) Copy content Toggle raw display
$59$ \( T^{6} - 104 T^{4} + 2708 T^{2} + \cdots - 4176 \) Copy content Toggle raw display
$61$ \( (T^{3} - 10 T^{2} + 2 T + 124)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 14 T^{2} - 256)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 104 T^{4} + 1680 T^{2} + \cdots - 7424 \) Copy content Toggle raw display
$73$ \( T^{6} - 204 T^{4} + 8176 T^{2} + \cdots - 1856 \) Copy content Toggle raw display
$79$ \( T^{6} - 149 T^{4} + 648 T^{2} + \cdots - 464 \) Copy content Toggle raw display
$83$ \( (T^{3} + 8 T^{2} - 154 T - 648)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 189 T^{4} + 664 T^{2} + \cdots - 464 \) Copy content Toggle raw display
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